Abstract

We study the resonances of the operator $P(h) = -\Delta_x + V(x) +
\varphi(hx)$. Here $V$ is a periodic potential, $\varphi$ a
decreasing perturbation and $h$ a small positive constant. We prove
the existence of shape resonances near the edges of the spectral bands
of $P_0 = -\Delta_x + V(x)$, and we give its asymptotic expansions in
powers of $h^{\frac12}$.